A simplicity criterion for finite groups (Q1357566)

For a finite group \(G\), denote by \(\pi(G)\) the set of all primes dividing \(|G|\), by \(\pi_e(G)\) the set of all element orders of \(G\) and by \(\psi(G)\) the number of composite numbers in \(\pi_e(G)\). By cleverly using the classification of simple groups and known facts about the prime graph components of simple groups, the authors prove the following surprising Theorem B: For every finite group \(G\) we have the inequality \(|\pi(G)|\leq\psi(G)+3\); moreover, if equality holds, then \(G\) must be a simple group. The second main result, Theorem A, classifies all finite simple groups \(G\) satisfying the equality \(|\pi(G)|=\psi(G)+3\).